Finite strain solutions in compressible isotropic elasticity

Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.     

Discontinuous solutions of the Navier-Stokes equations for compressible flow

We prove local existence and study properties of discontinuous solutions of the Navier-Stokes equations for one-dimensional, compressible, nonisentropic flow. We assume that, modulo a step function, the initial data is in L² and the initial velocity and density are in the space BV. We show that the velocity and the temperature become smoothed out in positive time, and that discontinuities in the density, pressure, and gradients of the velocity and temperature persist for all time. We also show that for stable gases these discontinuities decay exponentially in time, more rapidly for smaller viscosities.     

Stress solutions to the three-dimensional problem of elasticity

New representations of the stress tensor in the linear theory of elasticity and thermoelasticity are proposed. These representations satisfy the equilibrium equations and the strain compatibility equation. The stress tensor is expressed in terms of a harmonic tensor or a harmonic vector. The second boundary-value problem for an elastic half-space and an elastic layer is solved as an example __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 3–35, August 2006.     

Compact finite difference-Fourier spectral method for three-dimensional incompressible Navier-Stokes equations

A new compact finite difference-Fourier spectral hybrid method for solving the three dimensional incompressible Navier-Stokes equations is developed in the present paper. The fifth-order upwind compact finite difference schemes for the nonlinear convection terms in the physical space, and the sixth-order center compact schemes for the derivatives in spectral space are described, respectively. The fourth-order compact schemes in a single nine-point cell for solving the Helmholtz equations satisfied by the velocities and pressure in spectral space is derived and its preconditioned conjugate gradient iteration method is studied. The treatment of pressure boundary conditions and the three dimensional non-reflecting outflow boundary conditions are presented. Application to the vortex dislocation evolution in a three dimensional wake is also reported. The project supported by the National Natural Science Foundation of China     

On numerical solutions of the time-dependent Euler equations for incompressible flow

This paper presents finite element methods for the non-stationary Euler equations of a two dimensional inviscid and incompressible flow. For the time discretization, we compare numerical results obtained by the use of a leap-frog scheme and a semi-implicit scheme of order two.     

Semi-analytical three-dimensional elasticity solutions for generally laminated composite plates

Semi-analytical solutions for bending and free vibration of composite laminated plates have been derived based on three-dimensional elasticity theory using a newly developed hybrid analysis, which perfectly combines the state space approach (SSA) and the technique of differential quadrature (DQ). The thickness direction of laminates is selected as the transfer direction in SSA, and the DQ technique is employed to discretize the in-plane domains. This actualizes the transformation of the original partial differential equations into a state equation consisting of first-order ordinary differential equations. In particular, the use of DQ technique makes ease of the treatment of various boundary conditions, which cannot be considered in the conventional exact SSA. To avoid numerical instabilities in the conventional transfer matrix method, artificial interfaces are introduced to divide each layer into several sub-layers to reduce the transfer distance and the joint coupling matrices are established according to the continuity conditions at actual and artificial interfaces to implement the global analysis. Comprehensive numerical examples are preformed to validate the present hybrid method. Effects of some parameters on mechanical properties of the laminates are discussed.     

A class of exact solutions for N-dimensional incompressible magnetohydrodynamic equations

In this paper, a sufficient and necessary condition is presented for existence of a class of exact solutions to N-dimensional incompressible magnetohydrodynamic(MHD)equations. Such solutions can be explicitly expressed by appropriate formulae. Once the required matrices are chosen, solutions to the MHD equations are directly constructed.     

Solving the three-dimensional equations of the linear theory of elasticity

The three-dimensional Lamé equations are solved using Cartesian and curvilinear orthogonal coordinates. It is proved that the solution includes only three independent harmonic functions. The general solution of equations of elasticity for stresses is found. The stress tensor is expressed in both coordinate systems in terms of three harmonic functions. The general solution of the problem of elasticity in cylindrical coordinates is presented as an example. The three-dimensional stress–strain state of an elastic cylinder subjected, on the lateral surface, to arbitrary forces represented by a series of eigenfunctions is determined. An axisymmetric problem for a finite cylinder is solved numerically     

Functional and measure-valued solutions of the euler equations for flows of incompressible fluids

We consider the notion of a functional solution of the Euler equations for incompressible fluid flows. We show that a functional solution can be constructed under very weak a priori estimates on approximate solution sequences of the equation; an estimate uniform in L _loc ¹ together with weak consistency with the equation is sufficient to construct a solution. We prove that if we have an estimate uniform in L _loc ² available for the approximate solution sequence, then the structured functional solution just described becomes a measure-valued solution in the sense of DiPerna & Majda. We also show that a functional solution can be obtained from a measure-valued solution. We give an example showing that a much higher concentration of energy than in the case of measure-valued solutions is allowed by the approximation procedure of a functional solution.     

Group analysis and similarity solutions of the compressible boundary layer equations

Summary In this paper the application of Lie's methods to the equations of the laminar boundary layer is discussed. The momentum and energy equations in Prandtl's form are considered for a steady, viscous, compressible laminar flow with non zero pressure gradient, variable viscosity and thermal conductivity. Group analysis yields similarity solutions for given pressure distributions and particular values of the invariance group parameters (group classification). Crocco's transformation is obtained for the infinite-dimensional group of the Lie's algebra admitted by the equations.

---

Sommario In questa nota si applicano i metodi di Lie alle equazioni dello strato limite laminare nella forma di Prandtl per un fluido viscoso, compressibile, con gradiente di pressione non nullo e con viscosità e conducibilità termica variabili. L'analisi gruppale fornisce soluzioni di similarità per date distribuzioni di pressione e valori particolari dei parametri del gruppo di invarianza. La trasformazione di Crocco si ottiene in corrispondenza della parte infinito-dimensionale dell'algebra di Lie ammessa dalle equazioni.

     

Finite element stream function-vorticity solutions of the incompressible Navier-Stokes equations

The incompressible, two-dimensional Navier-Stokes equations are solved by the finite element method (FEM) using a novel stream function/vorticity formulation. The no-slip solid walls boundary condition is applied by taking advantage of the simple implementation of natural boundary conditions in the FEM, eliminating the need for an iterative evaluation of wall vorticity formulae. In addition, with the proper choice of elements, a stable scheme is constructed allowing convergence to be achieved for all Reynolds numbers, from creeping to inviscid flow, without the traditional need for upwinding and its associated false diffusion. Solutions are presented for a variety of geometries.     

An approach to three-dimensional elasticity problem of cylindrical orthotropic bodies

The state equation for a static or dynamic thermal elastic problem of cylindrical orthotropic bodies is derived using the three-dimensional theory of elasticity. The governing equation for displacement components is obtained by series expansion, and for simplicity the case of an axisymmetric problem of transverse isotropic bodies is given. For illustration, an axisymmetric problem of a combined-cylinder made of two transverse isotropic materials is also presented. Some interesting and significant results are obtained.Supported by-National Natural Science Foundation and Post-Doctor's Foundation of China     

Exact solutions of the equations of motion for an incompressible viscoelastic Maxwell medium

The unsteady plane-parallel motion of a incompressible viscoelastic Maxwell medium with constant relaxation time is considered. The equations of motion of the medium and the rheological relation admit an extended Galilean group. The class of solutions of this system which are partially invariant with respect to the subgroup of the indicated group generated by translation and Galilean translation along one of the coordinate axes is studied. The system does not have invariant solutions, and the set of partially invariant solutions is very narrow. A method for extending the set of exact solutions is proposed which allows finding solutions with a nontrivial dependence of the stress tensor elements on spatial coordinates. Among the solutions obtained by this method, the solutions describing the deformation of a viscoelastic strip with free boundaries is of special interest from a point of view of physics. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 2, pp. 16–23, March–April, 2009.     

Boundary integral equations of unique solutions in elasticity

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｂｏｕｎｄａｒｙｅｌｅｍｅｎｔｍｅｔｈｏｄ（ＢＥＭ）ｐｒｏｖｉｄｅｓａｎａｔｔｒａｃｔｉｖｅａｌｔｅｒｎａｔｉｖｅｆｏｒｔｈｅａｎａｌｙｓｉｓｏｆｅｎｇｉｎｅｅｒｉｎｇｐｒｏｂｌｅｍｓ．Ｉｔｓｍａｉｎａｄｖａｎｔａｇｅｓａｒｅｅｃｏｎｏｍｉｃａｌａｎｄｐａｒｔｉｃｕｌａｒｌｙｃｏｎｖｅｎｉｅｎｔｆｏｒｕｎｂｏｕｎｄｅｄｄｏｍａｉｎａｎｄｓｔｒｅｓｓｃｏｎｃｅｎｔｒａｔｉｏｎｐｒｏｂｌｅｍｓ．Ｔｈｅｂｏｕｎｄａｒｙｉｎｔｅｇｒａｌｅｑｕａｔｉｏｎ（ＢＩＥ）ｉｓｔｈｅ…     

Asymptotics of solutions of nonautonomous second-order ordinary differential equations close to linear equations

We find asymptotic representations for certain classes of solutions of nonautonomous second-order differential equations that are close, in a certain sense, to linear equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 2, pp. 230–241, April–June, 2008.     

Similarity solutions of unsteady laminar incompressible boundary layer equations for flow,heat and mass transfer in non-Newtonian fluids around axisymmetric bodies

Summary The possible existence of similarity solutions for the unsteady three-dimensional boundary layer flows with heat and mass transfer around stationary axisymmetric bodies which are fully immersed in purely viscous moving non-Newtonian fluids has been searched in general by the application of transformations, involving a single linear parameter. In particular, the cases involving rotational flows around stationary bodies and rotating bodies have been discussed as corollaries of the main analysis. The main analysis shows that the similarity solutions are possible only for the bodies for which where is a cross-sectional radius; and is the longitudinal distance from the nose point to the cross section. In case of rotating bodies, similarity solutions exist only for cones and disks. The analysis, as an example, has successfully been applied to the Powell-Eyring model. It is seen that for the same rate of shear, expenditure of energy for maintaining the rotation of the solid body is comparatively higher for a flow with a higher Eyring number where the Eyring numberEy=1/µBE. µ, B, andE are the material functions of the Powell-Eyring fluid.

---

Zusammenfassung Es wird die mögliche Existenz von Ähnlichkeitslösungen für instationäre drei-dimensionale Grenzschichtströmungen rein-viskoser nicht-newtonscher Flüssigkeiten mit Wärme- und Stoffübertragung um voll eingetauchte stationäre achsensymmetrische Körper in allgemeiner Weise untersucht. Hierbei werden Transformationen verwendet, die einen einzigen linearen Parameter enthalten. Als Spezialfälle der allgemeinen Analyse werden Rotationsströmungen um ruhende und rotierende Körper diskutiert. Die Hauptanalyse ergibt, daß Ähnlichkeitslösungen nur existieren für Körper mit , wo den Abstand von der Achse und den longitudinalen Abstand auf der Oberfläche von der Nase des Körpers aus bedeuten. Im Falle rotierender Körper existieren solche Lösungen nur für Kegel und Kreisscheiben. Die Analyse läßt sich erfolgreich auf das Beispiel einer Powell-Eyring-Flüssigkeit anwenden. Man findet, daß bei gleicher Schergeschwindigkeit der Energieverbrauch zur Aufrechterhaltung der Körperrotation mit wachsender Eyring-Zahl Ey= 1/µBE ansteigt, wobeiµ, B undE Materialfunktionen der Powell-Eyring-Flüssigkeit bedeuten.

---

---

With 1 figure and 1 table     

Finite element solutions of axisymmetric Euler equations for an incompressible and inviscid fluid

In this paper we present a finite element method for the numerical solution of axisymmetric flows. The governing equations of the flow are the axisymmetric Euler equations. We use a streamfunction angular velocity and vorticity formulation of these equations, and we consider the non-stationary and the stationary problems. For industrial applications we have developed a general model which computes the flow past an annular aerofoil and a duct propeller. It is able to take into account jumps of angular velocity and vorticiy in order to model the flow in the presence of a propeller. Moreover, we compute the complete flow around the after-body of a ship and the interaction between a ducted propeller and the stern. In the stationary case we have developed a simple and efficient version of the characteristics/finite element method. Numerical tests have shown that this last method leads to a very fast solver for the Euler equations. The numerical results are in good agreement with experimental data.